Monday, 27 June 2011

Last week I posted some lateral thinking and trick questions. The idea being that, when I do outreach workshops to younger audiences, they will be less worried about being wrong if they realise that everyone will be as wrong as they are. Again, I would like to reiterate the point that this is nothing about making the students feel stupid, but rather empowered as they should not be worried about make "silly" suggestions.

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The questions starts off with "You're driving a bus". Thus, the eye colour of the driver is whatever yours is.

The idea behind this question is to emphasise the point that to produce an answer you first have to understand what information you need from the question.

If you take 307 bananas from 429 bananas, how many bananas do you have? You have 307.

All months have 29 days.

Questions 2 and 3 deal with the idea of clarity of communication. When
answering a question in mathematics you have to clearly define your
terms and your assumptions.

The probability that exactly five are in the right envelopes is zero. If five are in the right envelopes the sixth must be two.

Here, again, we are dealing with the idea of taking the important information from the question. The most important word in the question is "exactly". Thus, any solution we generate should be weighed up against the requirement.

Each dog takes five days to dig a hole. So ten dogs will take five days to dig ten holes.

The full stop at the end of the sentence is the smallest circle.

By now the students should realise that the questions are trick
question, so questions 5 and 6 teaches them not to be too hasty with their
answer and to think carefully even when the answer appears obvious.

Tuesday, Thursday, today and tomorrow.

This question asks them to find a solution to the seemingly impossible. The idea being that mathematicians need tenacity when working with a problem. Many times you it will seem like the question is intractable but eventually you will find the right path.

Noah built the Ark not Moses.

All the numbers are divisible by two. The question does not ask for integer answers.

Questions 8 and 9, again, show the importance of reading questions carefully and fully
understanding what is being asked. Trust me, as a mathematician, the
hard part is not generating a solution, but rather, understanding what
the question is asking.

And the final puzzle. Did you spot it? The first instruction you are given is to write your name in the square. Next to this instruction is a rectangle. The square is at the bottom :).

Monday, 20 June 2011

Below are a few of my favourite lateral thinking and trick questions. Whenever I do outreach work shops on higher mathematics for secondary school I often open with this as a 5 minute ice breaker. Hopefully, the kids will not do very well. The reason I say hopefully is because I want to emphasise that they do not know everything. This is not meant in a bad way, but an encouraging way. Over the course of the workshop I touch on subjects such as logic and topology which they will have never seen. By getting the audience comfortable with being wrong and not knowing the answer I hope to encourage them to ask questions and provide solutions, even if they turn out to be wrong.

Fear of being wrong or asking "silly questions" is a common barrier to over come in a class room situation. The participants are surrounded by their peers and the last thing they want it to do is to seem stupid. However, the workshops flow better if everyone is willing to suggest their insights on which to build.

So, without further ado, try the questions yourself. Only give yourself 5 minutes to try out all the questions and, of course, don't cheat. I'll post the answers next Monday, so you can see how well you did. Remember

Research is what you do when you don't know what you are doing

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Write your name in the square:

You're driving a bus that is leaving on a trip from A and ending in B. To start off with, there were 32 passengers on the bus. At the next bus stop, 11 people get off and 9 people get on. At the next bus stop, 2 people get off and 2 people get on. At the next bus stop, 12 people get on and 16 people get off. At the next bus stop, 5 people get on and 3 people get off. Question: What colour are the bus driver's eyes?

If you take 307 bananas from 429 bananas, how many bananas do you have?

In a leap year how many months have 29 days?

A secretary prints out six different letters and is in a rush so she randomly stuffs the letters into six envelopes going to six different addresses. What is the probability that exactly 5 letters are in the right envelopes.

If five dogs dig five holes in five days, how long will it take ten dogs to dig ten holes?

Highlight, in some way, the smallest circle.

Name four days which start with the letter T.

If animals enter the Ark in pairs at rate of 30 pairs per day, how many days would it take Moses to get 360 individual animals on board?

How many numbers between zero and ten, inclusive, can be divided by two?

Monday, 13 June 2011

F set the coins out in a row
And chalked on each a letter, so,
To form the words "F AM NOT LICKED"
(An idea in his brain had clicked).
And now his mother he'll enjoin:
MA DO LIKE
ME TO FIND
FAKE COIN
-- Cedric A.B. Smith

Now this may appear to not make much sense, but it is in fact a really clever solutions to the 12 coin problem:

Imagine you are given 12 coins, and a set of weighing scales. 11 of the coins have the same weight, but one has a different weight. To make matters worse, you do not know if it is heavier of lighter than the others. The problem is to find out which coin is different, and whether it is lighter or heavier, using at most three
weighings on a pair of scales.

Before you see the answer and I explain the connection between the puzzle and the poem, have a go at solving the problem yourself. Here is a flash applet which allows you to play that game yourself:

Did you solve it? Did the poem help? If you managed to do it, give yourself a pat on the back. Otherwise, let me explain. Firstly, do as the poem says; on the coins write one of the single letters F, A, M, N, O, T, L, I, C, K, E and D. Now, we weigh the coins

MADO against LIKE

METO against FIND

FAKE against COIN

in each of the weighings we have two possibilities. Either, the pans balance or they don't. If any of the weighings do balance, we immediately know that all 8 coins are genuine and the dodgy coin is in the four we haven't weighed. Conversely, if the pans do not balance then we know that the four coins not on the scales are genuine.

As a specific example, suppose that, in our weighings, the right pan is always lower. We can then logically deduce that the coin "I" is the dodgy one and that it is heavier, since coin "I" is the only coin that appears in the right balance in all three weighings. If you are interested in the entire solution to this problem look at the last post in this Dr. Math forum post.

An idea not communicated can scarcely be said to exist.

I am a researcher of mathematical biology at the University of Oxford. Although I now do mathematics as a career I remember how hard maths was when I first started. I also remember what caused things to make sense. I try to relay these insights to everyone, with the hope that they, too, will understand.
Home page:
http://people.maths.ox.ac.uk/~woolley/index.htm